A semipositive holomorphic line bundle, in the smooth Hermitian sense, is one admitting a smooth Hermitian
metric with nonnegative real Chern curvature
form. This is a real 2-form of type 
. On a compact complex
manifold, smooth semipositivity implies analytic nefness,
while analytic nefness allows approximate lower
curvature bounds with metrics depending on the approximation parameter. A semipositive
singular Hermitian metric instead has
local weights that are plurisubharmonic
and may have unbounded singularities.
On a smooth complex projective variety, the existence of one such smooth Hermitian metric is stronger input than numerical nefness alone.
